Exact solutions to linear programming problems

The use of floating-point calculations limits the accuracy of solutions obtained by standard LP software. We present a simplex-based algorithm that returns exact rational solutions, taking advantage of the speed of floating-point calculations and attempting to minimize the operations performed in rational arithmetic. Extensive computational results are presented.

[1]  Thorsten Koch The final NETLIB-LP results , 2004, Oper. Res. Lett..

[2]  Heidelberg Interdisziplinäres Zentrum Für Wissenschaftliches Rechnen,et al.  [Optimierung von Materialien, Bauelementen, Schaltkreisen und Mikrosystemen : Simulation von Strömungsvorgängen in Mikro-Aggregaten ; Beginn: 01.01.2001, Ende: 31.12.2003] , 2004 .

[3]  R. Gomory AN ALGORITHM FOR THE MIXED INTEGER PROBLEM , 1960 .

[4]  Tommy Färnqvist Number Theory Meets Cache Locality – Efficient Implementation of a Small Prime FFT for the GNU Multiple Precision Arithmetic Library , 2005 .

[5]  Martin W. P. Savelsbergh,et al.  Lifted Cover Inequalities for 0-1 Integer Programs: Complexity , 1999, INFORMS J. Comput..

[6]  Daniel G. Espinoza On Linear Programming, Integer Programming and Cutting Planes , 2006 .

[7]  Christian Jansson,et al.  Rigorous Lower and Upper Bounds in Linear Programming , 2003, SIAM J. Optim..

[8]  Martin W. P. Savelsbergh,et al.  A Computational Study of Search Strategies for Mixed Integer Programming , 1999, INFORMS J. Comput..

[9]  Giovanni Rinaldi,et al.  An efficient algorithm for the minimum capacity cut problem , 1990, Math. Program..

[10]  N. Sloane,et al.  A linear programming bound for orthogonal arrays with mixed levels , 1996 .

[11]  Kurt Mehlhorn,et al.  Certifying and repairing solutions to large LPs how good are LP-solvers? , 2003, SODA '03.

[12]  Arnold Neumaier,et al.  Safe bounds in linear and mixed-integer linear programming , 2004, Math. Program..

[13]  Douglas H. Wiedemann Solving sparse linear equations over finite fields , 1986, IEEE Trans. Inf. Theory.

[14]  Gérard Cornuéjols,et al.  K-Cuts: A Variation of Gomory Mixed Integer Cuts from the LP Tableau , 2003, INFORMS J. Comput..

[15]  Bernd Gärtner,et al.  Exact arithmetic at low cost—a case study in linear programming , 1999, SODA '98.

[16]  William J. Cook,et al.  Implementing the Dantzig-Fulkerson-Johnson algorithm for large traveling salesman problems , 2003, Math. Program..

[17]  David S. Johnson,et al.  Data structures for traveling salesmen , 1993, SODA '93.

[18]  Martin W. P. Savelsbergh,et al.  Lifted Cover Inequalities for 0-1 Integer Programs: Computation , 1998, INFORMS J. Comput..